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Pricing European passport option with radial basis function. (English) Zbl 1397.91592
Summary: Passport option is a financial derivative with the value of a trading account as the underlying security. The valuation of this option can be obtained through the solution of a backward partial differential equation. A closed form solution for this valuation problem exists for the symmetric case when the risk-free rate is identical to the cost of carry. We obtain the value of the European passport option in this case, by using radial basis function and extend it to the asymmetric case when the risk-free rate is distinct from the cost of carry. The numerical schemes and algorithm are presented along with illustrative examples.

MSC:
91G60 Numerical methods (including Monte Carlo methods)
91G20 Derivative securities (option pricing, hedging, etc.)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Software:
Matlab; PATH Solver
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References:
[1] Hyer, T; Lipton-Lifschitz, A; Pugachevsky, D, Passport to success, Risk, 10, 127-131, (1997)
[2] Topper, J.: A finite element implementation of passport options. M.Sc. Thesis, University of Oxford (2003)
[3] Andersen, L; Andreasen, J; Brotherton-Ratcliffe, R, The passport option, J. Comput. Finance, 1, 15-36, (1998)
[4] Nagayama, I, Pricing of passport option, J. Math. Sci. Univ. Tokyo, 5, 747-785, (1998) · Zbl 0927.91011
[5] Chan, S.-S.: The valuation of American passport options. University of Wisconsin-Madison, Working Paper (1999)
[6] Dirkse, SP; Ferris, MC, The PATH solver: a non-monotone stabilization scheme for mixed complementarity problems, Optim. Methods Softw., 5, 123-156, (1995)
[7] Ahn, H; Penaud, A; Wilmott, P, Various passport options and their valuation, Appl. Math. Finance, 6, 275-292, (1999) · Zbl 1009.91038
[8] Penaud, A; Wilmott, P; Ahn, H, Exotic passport options, Asia Pacific Financ. Mark., 6, 171-182, (1999) · Zbl 1153.91553
[9] Henderson, V; Hobson, D, Local time, coupling and the passport option, Finance Stoch., 4, 69-80, (2000) · Zbl 0944.60046
[10] Henderson, V; Hobson, D, Passport options with stochastic volatility, Appl. Math. Finance, 8, 97-118, (2001) · Zbl 1013.91046
[11] Hull, J; White, A, The pricing of options on assets with stochastic volatilities, J. Finance, 42, 281-300, (1987) · Zbl 1126.91369
[12] Stein, EM; Stein, JC, Stock price distributions with stochastic volatility: an analytic approach, Rev. Financ. Stud., 4, 727-752, (1991) · Zbl 1458.62253
[13] Shreve, SE; Vecer, J, Options on a traded account: vacation calls, vacation puts and passport options, Finance Stoch., 4, 255-274, (2000) · Zbl 0997.91020
[14] Hajek, B, Mean stochastic comparison of diffusions, Zeitschrift fr Wahrscheinlichkeitstheorie und Verwandte Gebiete, 68, 315-329, (1985) · Zbl 0537.60050
[15] Pooley, D.: Numerical methods for nonlinear equations in option pricing. Ph.D. Thesis, University of Waterloo (2003)
[16] Baojun, B., Yang, W.: A viscosity solution approach to valuation of passport options in a jump-diffusion model. In: Proceedings of the 27th Chinese Control Conference, Kunming, Yunnan, China, pp. 606-608 (2008)
[17] Baojun, B; Yang, W; Jizhou, Z, Viscosity solutions of HJB equations arising from the valuation of European passport options, Acta Math. Sci., 30, 187-202, (2010) · Zbl 1224.91148
[18] Kampen, J, Optimal strategies of passport options, Math. Ind., 12, 666-670, (2008) · Zbl 1308.91164
[19] Malloch, H., Buchen, P.W.: Passport option: continuous and binomial models. Finance and Corporate Governance Conference (2011) · Zbl 1402.91756
[20] Chen, W., Fu, Z.-J., Chen, C.S.: Recent Advances in Radial Basis Function Collocation Methods. Springer Briefs in Applied Sciences and Technology. Springer-Verlag, Berlin, Heidelberg (2014) · Zbl 1282.65160
[21] Iske, A.: Scattered Data Modelling Using Radial Basis Functions, Tutorials on Multiresolution in Geometric Modelling. Springer, Berlin, Heidelberg, pp. 205-242 (2002) · Zbl 1004.65014
[22] Chen, S; Cowan, CFN; Grant, PM, Orthogonal least squares learning algorithm for radial basis function networks, IEEE Trans. Neural Netw., 2, 302-309, (1991)
[23] Liu, GR; Gu, YT, A point interpolation method for two-dimensional solids, Int. J. Numer. Methods Eng., 50, 937-951, (2001) · Zbl 1050.74057
[24] Liu, GR; Yan, L; Wang, JG; Gu, YT, Point interpolation method based on local residual formulation using radial basis functions, Struct. Eng. Mech., 14, 713-732, (2002)
[25] Liu, G.R., Gu, Y.T.: An Introduction to Meshfree Methods and Their Programming. Springer, Netherlands (2005)
[26] Rad, JA; Parand, K; Ballestra, LV, Pricing European and American options by radial basis point interpolation, Appl. Math. Comput., 251, 363-377, (2015) · Zbl 1328.91286
[27] Kumar, A; Tripathi, LP; Kadalbajoo, MK, A numerical study of Asian option with radial basis functions based finite differences method, Eng. Anal. Bound. Elem., 50, 1-7, (2015) · Zbl 1403.91373
[28] Buhmann, MD, Radial basis functions: theory and implementations, Camb. Monogr. Appl. Comput. Math., 12, 147-165, (2004)
[29] Fasshauer, G.E.: Meshfree Approximation Methods with Matlab, vol. 6. World Scientific, Singapore (2007) · Zbl 1123.65001
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